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Concept Explanation

Division of Polynomial by another Polynomial Long Division

Division of a polynomial by a binomial by using long division

For dividing a polynomial by a binomial, we may follow the following steps:

Step I Arrange the terms of the dividend and divisor in descending order of their degrees.

Step II Divide the first term of the dividend by the first term of the divisor to obtain the first term of the

quotient.

Step III Multiply the divisor by the first term of the quotient and subtract the result from the dividend to

obtain the remainder.

Step IV Consider the remainder (if any) as dividend and repeat step II to obtain the second term of

quotient.

Step V Repeat the above process till we obtain a remainder which is either zero or a polynomial of

degree less than that of the divisor.

Example 1 Divide $large 6+x-4x^{2} + x^{3};;by;;x-3.$

Solution We go through the following steps to perform the division:

Step I We write the terms of the dividend as well as of divisor in descending order of their

degrees. Thus, we write

$large dpi{120} large 6+x-4x^{2}+x^{3};;as;;x^{3}-4x^{2}+x+6;;and;;x-3;;as;;x-3$

Step II We divide the first term $large x^{3}$ of the dividend by the first term $large x$ of the divisor and obtain

$large frac{x^{3}}{x}=x^{2}$ as the first term of the quotient.

Step III We multiply the divisor $large x-3$ by the first term $large x^2$ we get $large x^3-3x^2$ of the quotient

and subtract the result from the dividend

$large x^{3}-4x^{2}+x+6.$

We obtain $large -x^{2}+x+6$ as the remainder.

Step IV We take $large -x^{2}+x+6$ as the new dividend and repeat step II to obtain the second

term $large (-frac{x^{2}}{x}=)-x$ of the quotient.

Step V We multiply the divisor $large x-3$ by the second term $large -x$ of the quotient and subtract the

result $large -x^{2} + 3x$ from the new dividend. We obtain $large -2x + 6$ as the remainder.

Step VI Now we treat $large -2x + 6$ as the new dividend and divide its first term $large -2x$ by the first

term $large x$ of the divisor to obtain $large frac{-2x}{x}=-2$ as the third term of the quotient.

Step VII We multiply the divisor $large x-3$ and the third term $large -2$ of the quotiemt and subtract the

result $large -2x+6$ from the new dividend. We obtain $large 0$ as the remainder.

Thus, we can say that

$large (6+x-4x^{2}+x^{3})div (x-3)=x^{2}-x-2$

or, $large frac{6+x-4x^{2}+x^{3}}{x-3}=x^{2}-x-2$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Divide and find the remainder: $frac{3x^3 + 4x^2 - 7x - 5}{3x - 2}$
A. - 4	B. - 7	C. 10	D. 14
Right Option : B
View Explanation

Question : 2

Find the remainder when $large 2x^4-3x^3+x^2+1$ is divided by x - 2.
A. 13	B. 14	C. 15	D. 16
Right Option : A
View Explanation

Question : 3

The process of division of a polynomial by another polynomial does not stop until ____________________
A. The degree of the remainder becomes less than that of the divisor		B. The remainder becomes zero
C. Both A and B		D. Neither A nor B
Right Option : C
View Explanation